Let [math]U[/math] be a uniformly distributed random variable on [math][0,1][/math]. What is the probability that the equation

has two distinct real roots [math]x_1[/math] and [math]x_2[/math]?

Write a program to simulate the random variables whose densities are given by the following, making a suitable bar graph of each and comparing the exact density with the bar graph.

• [math]f_X(x) = e^{-x}\ \ \mbox{on}\,\, [0,\infty)\,\, (\mbox{but\,\,just\,\,do\,\,it\,\,on\,\,} [0,10]).[/math]
• [math]f_X(x) = 2x\ \ \mbox{on}\,\, [0,1].[/math]
• [math]f_X(x) = 3x^2\ \ \mbox{on}\,\, [0,1].[/math]
• [math]f_X(x) = 4|x - 1/2|\ \ \mbox{on}\,\, [0,1].[/math]

Suppose we are observing a process such that the time between occurrences is exponentially distributed with [math]\lambda = 1/30[/math] (i.e., the average time between occurrences is 30 minutes). Suppose that the process starts at a certain time and we start observing the process 3 hours later. Write a program to simulate this process. Let [math]T[/math] denote the length of time that we have to wait, after we start our observation, for an occurrence. Have your program keep track of [math]T[/math]. What is an estimate for the average value of [math]T[/math]?

Jones puts in two new lightbulbs: a 60 watt bulb and a 100 watt bulb. It is claimed that the lifetime of the 60 watt bulb has an exponential density with average lifetime 200 hours ([math]\lambda = 1/200[/math]). The 100 watt bulb also has an exponential density but with average lifetime of only 100 hours ([math]\lambda = 1/100[/math]). Jones wonders what is the probability that the 100 watt bulb will outlast the 60 watt bulb.

If [math]X[/math] and [math]Y[/math] are two independent random variables with exponential densities [math]f(x) = \lambda e^{-\lambda x}[/math] and [math]g(x) = \mu e^{-\mu x}[/math], respectively, then the probability that [math]X[/math] is less than [math]Y[/math] is given by

where [math]G(x)[/math] is the cumulative distribution function for [math]g(x)[/math]. Explain why this is the case. Use this to show that [math] P(X \lt Y) = \frac \lambda{\lambda + \mu} [/math] and to answer Jones's question.

Consider the simple queueing process of Example. Suppose that you watch the size of the queue. If there are [math]j[/math] people in the queue the next time the queue size changes it will either decrease to [math]j - 1[/math] or increase to [math]j + 1[/math]. Use the result of Exercise to show that the probability that the queue size decreases to [math]j - 1[/math] is [math]\mu/(\mu +\lambda)[/math] and the probability that it increases to [math]j + 1[/math] is [math]\lambda/(\mu + \lambda)[/math]. When the queue size is 0 it can only increase to 1. Write a program to simulate the queue size. Use this simulation to help formulate a conjecture containing conditions on [math]\mu[/math] and [math]\lambda[/math] that will ensure that the queue will have times when it is empty.

Let [math]X[/math] be a random variable having an exponential density with parameter [math]\lambda[/math]. Find the density for the random variable [math]Y = rX[/math], where [math]r[/math] is a positive real number.

Let [math]X[/math] be a random variable having a normal density and consider the random variable [math]Y = e^X[/math]. Then [math]Y[/math] has a log normal density. Find this density of [math]Y[/math].

Let [math]X_1[/math] and [math]X_2[/math] be independent random variables and for [math]i = 1, 2[/math], let [math]Y_i = \phi_i(X_i)[/math], where [math]\phi_i[/math] is strictly increasing on the range of [math]X_i[/math]. Show that [math]Y_1[/math] and [math]Y_2[/math] are independent. Note that the same result is true without the assumption that the [math]\phi_i[/math]'s are strictly increasing, but the proof is more difficult.